Base (topology)

In topology, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.

Table of contents

1 Simple properties of bases
2 Objects defined in terms of bases
3 Theorems

Simple properties of bases

Two important properties of bases which together form an alternate definition are:

The base elements cover X.
Let B₁, B₂ be base elements and let I be their intersection. Then for each x in I, there is another base element B₃ containing x and contained in I.

If a collection fails to satisfy either of these, it is not a base for any topology; at best, it is a subbase. A sufficient but not necessary condition is that the base is closed under intersections; then we can always take B₃ = I above.

If we are given a topological space, we can verify whether or not some collection of open sets is a base for the space either using the above or directly from the definition. For example, given the standard topology on the real numbers, we know the open intervals are open. In fact they are a base, because the intersection of any two open intervals is itself an open interval or empty.

However, a base is not unique. Many bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the real numbers, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals. In contrast to the basis of linear algebra, the elements of a base need not be independent; that is, it may be possible to write one as a union of some of the others. In fact, any open sets in the space generated by a base may be safely added to the base without changing the topology.

An example of a collection of open sets which is not a basis is the set S of all semi-infinite intervals of the forms (−∞,a) and (a,∞), where a is a real number. Then S is not a base for any topology on R. To show this, suppose it were. Then, for example, (−∞,1) and (0,∞) would be in the topology generated by S, being unions of a single base element, and so their intersection (0,1) would be as well. But (0,1) clearly cannot be written as a union of the elements of S. Using the alternate definition, the second property fails, since no base element can "fit" inside this intersection.

Objects defined in terms of bases

The order topology is usually defined as the topology generated by a collection of open-interval-like sets.
The metric topology is usually defined as the topology generated by a collection of open balls.
A space is said to be second countable if it has a countable base.

Theorems

For each point x in an open set U, there is a base element containing x and contained in U.

A topology T₂ is finer than a topology T₁ if and only if for each x and each base element B of T₁ containing x, there is a base element of T₂ containing x and contained in B.
If B₁,B₂,...,B_n are bases for the topologies T₁,T₂,...,T_n, then the set product B₁ × B₂ × ... × B_n is a base for the topology '\'T₁ × T₂ × ... × Tn''. In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.
Let B is a base for X and let Y be a subspace of X. Then if we intersect each element of B with Y, the resulting collection of sets is a basis for the subspace Y.
If a function f:X → Y maps every base element of X into an open set of Y, it is an open map. Similarly, if every preimage of a base element of Y is open in X, then f is continuous.
A topology is a base which generates itself.

See also:

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